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Phasor Analysis of Pulse Tube Refrigerator Phasor Analysis of Pulse Tube Refrigerator L. Mohanta, M.D. Atrey Department of Mechanical Engineering Indian Institute of Technology Bombay, Powai Mumbai-400076, India ABSTRACT A phasor diagram for a pulse tube refrigerator (PTR) is a vectorial representation of mass flow rate, pressure, and temperature at different locations as a function of time. With the help of a phasor diagram, the operation of different types of pulse tube refrigerators can be well understood. Phasor analysis based on these diagrams gives an idea regarding the underlying complex phenomena of the PTR. In the present work, a simplified model has been presented based on the assumption that there is no phase difference between temperature and pressure throughout the working space. The phasor analysis is extended to a two-stage Orifice Pulse Tube Refrigerator (OPTR) and to a Double Inlet PTR (DIPTR). The important contribution of the work is that it highlights the condition for which the DIPTR will work better than the OPTR. INTRODUCTION In recent years, pulse tube refrigerator (PTR) technology has undergone a lot of improvement. PTRs are now widely used for various applications, e.g. MRI, NMR, space technology, cooling of SQUIDS etc. This is due to the absence of cold moving part in PTRS, which improves their vibration characteristics and reliability. The PTR works in a cyclic manner which consists of a) Compression, b) Constant Pressure Regenerative cooling, c) Expansion and d) Constant pressure regenerative heating [1]. Different analyses have been presented in order to analyse the underlying phenomena taking place in the PTR. Various numerical models have been proposed that give quite good result [2, 3]. However, for a DIPTR or for a two-stage PTR, these analyses become more mathematical in nature and therefore are not easy to understand. Also, they do not help us to imagine the physical processes that are taking place in the PTR. In view of this, the present work highlights the working of different types of PTR using phasor analysis, which is a vectorial representation of the working of the PTR. Basic analysis of the PTR using phasor diagrams was introduced by Radebaugh [4], while Kittel et al. [5] carried out similar analyses using Taylor series approximation. Hofmann and Pan [1] explained the working of different types of PTR with phasor diagrams and analyzed them using an electrical analogy and complex geometry. A phasor diagram for a Double Inlet PTR was suggested by Chokhawala et al. [6]. In the present work, a detailed phasor analysis for a two-stage PTR, both OPTR and DIPTR, is presented based on the law of conservation of mass. The phasor analysis helps to understand the importance of phase difference between mass flow rate at the cold end and the pressure pulse in the pulse tube. The refrigerating effect for different types of PTR strongly depends on the 299 300 PULSE TUBE ANALYSIS AND EXPERIMENTAL MEASUREMENTS phase shift arrangement and also on the phase difference. Due to the complexities of the flow involved in the two-stage DIPTR and OPTR, the phasor diagram put forth in the present work will prove to be very helpful to understand the importance of these parameters. The present work also introduces the concept and importance of phase lead of the double inlet flow entry in the DIPTR. The analysis also comes out with a necessary condition on the value of the phase lead angle in order to make a DIPTR function better in terms of refrigerating effect, as compared to an OPTR ANALYSIS FOR OPTR Assumptions a) The gas obeys the ideal gas law. b) There is no phase difference between the pressure and the mass flow rate at the hot end through the orifice of the pulse tube. c) There is no phase difference between the pressure and the temperature throughout the working space of the PTR. d) The variation of pressure, temperature and mass flow at a given location is sinusoidal. e) The pulse tube behaves adiabatically during the operation. Pressure and Temperature Variations In the pulse tube, pressure and temperature variations are given as: PPPcos( t ) 01 Z (1) TTT 01cos(Z t ) (2) Applying the adiabatic condition to the pulse tube, J 1 TP§·J ¨¸ TP 00©¹ (3) Using Taylor series expansion and neglecting higher order terms [5], T1 J 1 P1 (4) TP00J Mass Flow in the Pulse Tube Considering the whole pulse tube as a control volume and applying the law of conservation of mass, dV U pt UvvU (5) dt c h P Assuming that the pulse tube is at a mean temperature, density can be taken as U RTmpt dPV pt mmc h (6) RTmpt dt Vpt dP mmc h (7) RTmpt dt Mass flow rate at the hot end,m h , through orifice is directly proportional to pressure difference [5]. 'PP 1 cosZ t (8) Mass flow rate at hot end is given as [5], PHASOR ANALYSIS OF PULSE TUBE REFRIGERATOR 301 Th mCP h 11cosZt (9) Tc Combining Eq. (9) and Eq. (7), Th ZVpt mCPt c 11cosZZ Pt1cos( S / 2) (10) TRcmTpt From Eq (10), it is clear that mass flow rate at the cold end consists of two components. The first term in the right hand side is the contribution resulting from the mass flow rate at the hot end through the orifice which is in phase with the pressure. The second term represents the additional amount of mass entering the pulse tube due to pressurization, which is leading the earlier term by 90o. Fig.1 shows the equivalent phasor representation of Eq. (10). The phase angle ï represents the phase difference between the pressure pulse and T m c T. The numerical analysis of the first law of thermodynamics applied to the PTR also shows a similar phase difference in the mass flow rate at different locations of the pulse tube [7]. Enthalpy flow Due to the pulsating nature of the flow in the pulse tube, time-averaged enthalpy is considered for energy balance for a cycle. Fig. 2 shows enthalpy flow in the pulse tube. Applying the first law of thermodynamics to the cold end heat exchanger of the PTR as shown in Fig. 2 gives, Q ¢HH ²¢r ² (11) where Q is the refrigerating effect, ¢²H is the time-averaged enthalpy from the cold end heat exchanger of the pulse tube and ¢Hr ² is the time-averaged enthalpy flow from regenerator to cold end heat exchanger. For perfect regeneration, ¢²Hr =0 Hence the refrigerating effect obtained will be, C W Q ¢Hm ² p Tdt (12) ³ c W 0 Substituting the expression for mass flow rate from Eq. (10) and using Eq. (3), C W ¢²HmTT p ()( cos)Ztdt (13) ³ cc 1 W 0 Figure 1. Phasor diagram showing mass flow rate at hot end and cold end of pulse tube. 302 PULSE TUBE ANALYSIS AND EXPERIMENTAL MEASUREMENTS Figure 2. Enthalpy flow in the pulse tube. The integration of the second term in Eq. (10) becomes zero as this term is 900 out of phase. This shows there is no contribution to the refrigerating effect by the mass flow due to pressurization. Hence Eq. (13) reduces to C W T HCPtTT p h cos *( costdt ) (14) ¢² ³ 11 ZZc 1 W 0 Tc On further integration, C P2 p 1 ¢²H TCh 1 (15) 5 P0 This can be expressed in terms of mass flow rate at the cold end, m c . From Fig.1 m h and m c can be related as follows. m h 1 P1 cosT And QH ¢ ² RTmcc cosT (16) m c 2 P0 This result is in agreement with Radebaugh [4] and Kittel [5]. Eqs. (15), (16) show that the smaller is the phase difference, ï , and larger is the mass flow rate, m c , and the more refrigerating effect will be obtained. In other words, the product m c cosT has to be as large as possible. In the case of BPTR, the mass flow rate at the hot end ( m h ) is zero, which implies ï is equal to 900. Hence, theoretically, the refrigerating effect should be equal to zero for this configu- ration. However, due to heat transfer from gas to the wall of the pulse tube, ï decreases slightly below 900. The numerical value of ï could also be indicative of the losses in the system. As the loss increases, ï increases and that will lead to decrease in the refrigerating effect. PHASOR DIAGRAM FOR TWO STAGE OPTR The phasor diagram for a two-stage OPTR can be drawn in a similar fashion as that of the single-stage OPTR. For the two-stage PTR, both the pulse tubes are considered separately. Fig. 3 shows the phasor diagram for both stages of the pulse tube. Fig. 3(a) gives the phasor diagram for the second stage pulse tube of the two-stage OPTR where m h2 and m c2 represent the phasors for mass flow rate at the second stage hot and cold ends, respectively. It can be seen Figure 3. Phasor diagram for pulse tube (a) Second stage (b) First stage. PHASOR ANALYSIS OF PULSE TUBE REFRIGERATOR 303 from Fig. 3(a) that the m , phasor for mass flow rate at the hot end of the second stage Reg 2h ZP VReg 2 regenerator, can be obtained by adding the phasor for regenerator-2 loss, ( 1 ), to the RTmgRe 2 phasor for mass flow rate at the cold end of the pulse tube-2,m c2 Fig. 3(b) gives the phasor diagram for the first stage of the two stage OPTR. The flow coming out of regenerator-1 gets divided into two streams; one stream goes to the first stage pulse tube and the second one goes through regenerator-2.
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